Senate Republicans likely will need to invoke the “nuclear option” — changing Senate rules to remove the filibuster option — to elevate Gorsuch to the high court after it became clear on Monday that all but a handful of Democrats would support a filibuster against the 49-year-old Coloradan.

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But debate over Senate procedure may be too arcane to resonate beyond the Beltway. The end result — Gorsuch’s eventual confirmation — is what matters, experts say.

The imminent win is drawing sighs of relief from GOP insiders given the president’s struggles so far. Trump’s approval ratings are historically low for such a new president.

The latest Gallup tracking poll, released Monday, showed him being disapproved of by 57 percent of adults. Only 38 percent approved of his job performance.

Those numbers are partly a consequence of the polarized political landscape and the ongoing controversy into alleged Russian meddling in last year’s presidential election.

But they have surely not been helped by the debacle of Trump’s failed push to repeal and replace the Affordable Care Act. Another of the president’s major initiatives, his temporary travel ban focusing on six predominantly Muslim countries, sparked street protests and is bogged down in the courts.

Putting a conservative in the Supreme Court seat once held by Justice Antonin Scalia offers some respite.

“The administration needs somewhere where they are able to say, ‘We have accomplished an outcome that the president wanted,’ ” said Grant Reeher, a political science professor at Syracuse University.

Gorsuch’s confirmation is unlikely to represent a transformative breakthrough for Trump, however.

Reeher described it as “a normal accomplishment — and I think he needs a couple of those at this point, given everything else.”

Republicans, especially those beyond the circle of hardcore Trump loyalists, argue there is another lesson to be learned. An administration that has often seemed to take pleasure in thumbing its nose at Washington convention played largely by the rules in its quest to replace Scalia — and it has worked.

The process began during the campaign, when Trump released a list of 21 possible replacements for Scalia, based mostly on input from conservative groups such as the Federalist Society and the Heritage Foundation.

The whole process was starkly at odds with the travel ban, which was widely criticized as too slapdash, as well as the panic to round up votes on the effort to repeal ObamaCare.

“It offers a real playbook for them going forward. They took a deliberative, substantive, thoughtful process,” said GOP strategist and Hill contributor Matt Mackowiak. “To me, that’s another thing here — not just that it looks normal, but that it also offers a case study for the White House to try to replicate in other situations.”

Still, even Mackowiak acknowledged that Trump was likely to get only a small polling bump at best in the wake of Gorsuch’s likely confirmation.

There is no certainty that the administration’s probable Supreme Court success will overshadow the other problems that are crowding in upon the White House. The story of Russian election interference, and the possibility of links between Moscow and people in Trump’s orbit, is Exhibit A when it comes to the depth of those challenges.

New details related to that story can erupt at any moment, as was proved once again on Monday afternoon when The Washington Post reported that an attempt had been made in January to establish an apparent back channel between Trump and Russia.

According to the Post, a meeting had been facilitated by the United Arab Emirates between Erik Prince — the controversial founder of Blackwater and the brother of Trump’s Education secretary, Betsy DeVos — and a Russian close to Vladimir Putin, the nation’s president.

Speaking to The Hill shortly before that story emerged, Democratic strategist Tad Devine said the positive effects of a Supreme Court success for the president could be limited because of his other travails.

“There is so much flak flying around with Trump right now,” Devine said. “Until the flak goes away … I don’t think anything in the short term is going to remedy his situation. He started off with not-great standing and he has made it worse.”

“I think the Democrats really had to filibuster this,” he said. “If they hadn’t, it would have been a tremendous loss for them, especially with their base. The base would have been inconsolable, frankly.”

Republican hopes that they could persuade eight members of the minority to vote to end debate on Gorsuch’s nomination, and so thwart a filibuster, were dashed in the end.